1.

If α, β be the two roots of the equation x2 + x + 1 = 0, then the equation whose roots are \(\frac{α}{β}\) and \(\frac{β}{α}\) is ?(a) x2 – x – 1 = 0 (b) x2 – x + 1 = 0 (c) x2 + x – 1 = 0 (d) x2 + x + 1 = 0

Answer»

(d) x2 + x + 1 = 0

Let α, β be the roots of the equations x2 + x + 1 = 0. Then, 

Sum of roots = α + β = – 1, Product of roots = αβ = 1 

Now the equation whose roots are \(\frac{α}{β}\) and \(\frac{β}{α}\) is

x2 - \(\big(\frac{α}{β} + \frac{β}{α}\big)x\) + \(\big(\frac{α}{β} \times \frac{β}{α}\big)=0.\)

\(\frac{α}{β}\) + \(\frac{β}{α}\) = \(\frac{α^2+β^2}{αβ}\) = \(\frac{(α+β)^2-2αβ}{αβ}\) = \(\frac{(-1)^2-2(1)}{1}\) = -1 and \(\frac{α}{β}\) x \(\frac{β}{α}\) = 1.

∴ Required equation = x2 + x + 1 = 0.


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